A base-p Sprague–Grundy-type theorem for p-calm subtraction games: Welter’s game and representations of generalized symmetric groups
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Yuki Irie
Abstract
For impartial games Γ and Γ′, the Sprague-Grundy function of the disjunctive sum Γ + Γ′ is equal to the Nim-sum of their Sprague-Grundy functions. In this paper, we introduce p-calm subtraction games and show that for p-calm subtraction games Γ and Γ′, the Sprague-Grundy function of a p-saturation of Γ+Γ′ is equal to the p-Nim-sum of the Sprague-Grundy functions of their p-saturations. Here a p-Nim-sum is the result of addition without carrying in base p, and a p-saturation of Γ is an impartial game obtained from Γ by adding some moves. It will turn out that Nim and Welter’s game are p-calm. Further, using the p-calmness of Welter’s game, we generalize a relation between Welter’s game and representations of symmetric groups to disjunctive sums of Welter’s games and representations of generalized symmetric groups; this result is described combinatorially in terms of Young diagrams.
Abstract
For impartial games Γ and Γ′, the Sprague-Grundy function of the disjunctive sum Γ + Γ′ is equal to the Nim-sum of their Sprague-Grundy functions. In this paper, we introduce p-calm subtraction games and show that for p-calm subtraction games Γ and Γ′, the Sprague-Grundy function of a p-saturation of Γ+Γ′ is equal to the p-Nim-sum of the Sprague-Grundy functions of their p-saturations. Here a p-Nim-sum is the result of addition without carrying in base p, and a p-saturation of Γ is an impartial game obtained from Γ by adding some moves. It will turn out that Nim and Welter’s game are p-calm. Further, using the p-calmness of Welter’s game, we generalize a relation between Welter’s game and representations of symmetric groups to disjunctive sums of Welter’s games and representations of generalized symmetric groups; this result is described combinatorially in terms of Young diagrams.
Chapters in this book
- Frontmatter I
- Preface V
- Contents XIII
- The game of flipping coins 1
- The game of blocking pebbles 17
- Transverse Wave: an impartial color-propagation game inspired by social influence and Quantum Nim 39
- A note on numbers 67
- Ordinal sums, clockwise hackenbush, and domino shave 77
- Advances in finding ideal play on poset games 99
- Strings-and-Coins and Nimstring are PSPACE-complete 109
- Partizan subtraction games 121
- Circular Nim games CN(7, 4) 139
- Misère domineering on 2 × n boards 157
- Relator games on groups 171
- Playing Bynum’s game cautiously 201
- Genetically modified games 229
- Game values of arithmetic functions 245
- A base-p Sprague–Grundy-type theorem for p-calm subtraction games: Welter’s game and representations of generalized symmetric groups 281
- Recursive comparison tests for dicot and dead-ending games under misère play 309
- Impartial games with entailing moves 323
- Extended Sprague–Grundy theory for locally finite games, and applications to random game-trees 343
- Grundy numbers of impartial three-dimensional chocolate-bar games 367
- On the structure of misère impartial games 389
Chapters in this book
- Frontmatter I
- Preface V
- Contents XIII
- The game of flipping coins 1
- The game of blocking pebbles 17
- Transverse Wave: an impartial color-propagation game inspired by social influence and Quantum Nim 39
- A note on numbers 67
- Ordinal sums, clockwise hackenbush, and domino shave 77
- Advances in finding ideal play on poset games 99
- Strings-and-Coins and Nimstring are PSPACE-complete 109
- Partizan subtraction games 121
- Circular Nim games CN(7, 4) 139
- Misère domineering on 2 × n boards 157
- Relator games on groups 171
- Playing Bynum’s game cautiously 201
- Genetically modified games 229
- Game values of arithmetic functions 245
- A base-p Sprague–Grundy-type theorem for p-calm subtraction games: Welter’s game and representations of generalized symmetric groups 281
- Recursive comparison tests for dicot and dead-ending games under misère play 309
- Impartial games with entailing moves 323
- Extended Sprague–Grundy theory for locally finite games, and applications to random game-trees 343
- Grundy numbers of impartial three-dimensional chocolate-bar games 367
- On the structure of misère impartial games 389
